Question : A clock tower stands at the crossing of two roads which point in the north-south and the east-west directions. $P, Q, R$, and $S$ are points on the roads due north, east, south, and west respectively, where the angles of elevation of the top of the tower are respectively, $\alpha, \beta, \gamma$ and $\delta$. Then $\left(\frac{\mathrm{PQ}}{\mathrm{RS}}\right)^2$ is equal to:
Option 1: $\frac{\tan ^2 \alpha+\tan ^2 \beta}{\tan ^2 \gamma+\tan ^2 \delta}$
Option 2: $\frac{\cot ^2 \alpha+\cot ^2 \beta}{\cot ^2 \gamma+\cot ^2 \delta}$
Option 3: $\frac{\cot ^2 \alpha+\cot ^2 \delta}{\cot ^2 \beta+\cot ^2 \gamma}$
Option 4: $\frac{\tan ^2 \alpha+\tan ^2 \delta}{\tan ^2 \beta+\tan ^2 \gamma}$
Correct Answer: $\frac{\cot ^2 \alpha+\cot ^2 \beta}{\cot ^2 \gamma+\cot ^2 \delta}$
Solution : Let K be the point on the top of the tower and the height of the clock tower $OK$ be $h$ cm $OK$ is perpendicular to $PR$ and $SQ$. In Δ$POK$, $\tan α = \frac{OK}{OP}$ ⇒ $OP = \frac{h}{\tan α}$ ⇒ $OP = h \cot α$ Similarly, In ΔQOK $\tan β = \frac{OK}{OQ}$ ⇒ $OQ = \frac{h}{\tan β}$ ⇒ $OQ = h \cot β$ In $ΔPOQ$ formed by joining $P$ and $Q$, $OP$ is perpendicular to $OQ$, and then ⇒ $PQ^2 = OP^2 + OQ^2$ ⇒ $PQ^2 = h^2 \cot^2 α + h^2 \cot^2 β$ ⇒ $PQ^2 = h^2 (\cot^2 α + \cot^2 β)$ Similarly, $RS^2 = h^2 (\cot^2 γ + \cot^2 δ)$ ⇒ $(\frac{PQ}{RS})^2 = \frac{h^2 (\cot^2 α + \cot^2 β)}{h^2 (\cot^2 γ + \cot^2 δ)}$ ⇒ $(\frac{PQ}{RS})^2 = \frac{\cot^2 α + \cot^2 β}{\cot^2 γ + \cot^2 δ}$ Hence, the correct answer is $\frac{\cot^2 α + \cot^2 β}{\cot^2 γ + \cot^2 δ}$.
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Question : A pole stands vertically on a road, which goes in the north-south direction. P and Q are two points towards the north of the pole, such that $P Q=b$, and the angles of elevation of the top of the pole at $P, Q$ are $\alpha, \beta$ respectively. Then the height of the pole is:
Option 1: $\frac{b}{\tan \beta+\tan \alpha}$
Option 2: $\frac{b}{\tan \beta-\tan \alpha}$
Option 3: $\frac{b}{\cot \beta-\cot \alpha}$
Option 4: $\frac{\mathrm{b} \tan \alpha}{\tan \beta}$
Question : The angles of elevation of a pole from two points which are 75 m and 48 m away from its base are $\alpha$ and $\beta$, respectively. If $\alpha$ and $\beta$ are complementary, then the height of the tower is:
Option 1: 54.5 m
Option 2: 61.5 m
Option 3: 60 m
Option 4: 50 m
Question : Which of the following has the highest penetrating power?
Option 1: Alpha ray
Option 2: Beta ray
Option 3: Gamma ray
Option 4: Delta ray
Question : If $\sin \beta=\frac{1}{3},(\sec \beta-\tan \beta)^2$ is equal to:
Option 1: $\frac{3}{4}$
Option 2: $\frac{2}{3}$
Option 3: $\frac{1}{2}$
Option 4: $\frac{1}{3}$
Question : If $\sec A - \tan A = p$, then find the value of $\sec A$.
Option 1: $\frac{p^2-1}{p^2+1}$
Option 2: $\frac{\mathrm{p}^2+1}{\mathrm{p}^2-1}$
Option 3: $\frac{\mathrm{p}^2+1}{\mathrm{p}}$
Option 4: $\frac{\mathrm{p}^2+1}{2 \mathrm{p}}$
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